# Interacting Brownian Swarms: Some Analytical Results

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## Abstract

**:**

## 1. Introduction

## 2. Flocking of Interacting Brownian Agents

- (i)
- In real-time, agent ${\mathcal{A}}_{i}$ counts the (time-dependent) number ${N}_{U,i}\left(t\right)$ of leading fellows located within an observation interval U, namely the number of ${\mathcal{A}}_{k}$ for $k\ne i$ that are located in ${R}_{U,i}\left(t\right)$ with:$${R}_{U,i}\left(t\right):=\{x\in \mathbb{R}\phantom{\rule{0.166667em}{0ex}}\mid \phantom{\rule{0.166667em}{0ex}}0<x-{X}_{i}\left(t\right)\le U\}.$$
- (ii)
- The ${\mathcal{A}}_{i}$-rank-based drift in Equation (1) is then determined by:$${d}_{U,i}(\overrightarrow{X}\left(t\right),t):=\alpha \frac{{N}_{U,i}\left(t\right)}{N},\phantom{\rule{1.em}{0ex}}i=1,2,\cdots ,N,$$

#### Rank-Based Brownian Motions (RBMs)

## 3. Heuristic Characterization of the Observation Threshold Leading to Cooperative Dynamics

**Figure 1.**End probability distribution $P(x,T)$ at time $T=200>{\tau}_{\mathrm{relax}}=\frac{{\mu}_{s}}{{\sigma}^{2}}=\frac{13.58}{{1}^{2}}$ of swarms of $N=500$ agents, with respect to U. Here, all agents initially start at ${x}_{0}=-15$. (

**a**): tight swarm for $U=3{U}_{c}=3{\mu}_{S}$ (red) and “diffusive evaporating” swarm for $U=\frac{{U}_{c}}{3}$ (blue). (

**b**): tight swarm for $U=2{U}_{c}$ (red) and “diffusive evaporating” swarm for $U=\frac{{U}_{c}}{2}$ (blue). The respective observation ranges U are depicted at the top of each figure.

**Figure 2.**End probability distribution $P(x,T)$ at time $T={\tau}_{\mathrm{relax}}=13.58$ of a swarm of $N=500$ agents, initially starting at ${x}_{0}=-15$. The observation range $U={U}_{c}={\mu}_{S}$ is depicted at the top of the figure. Notice that the evaporation (i.e., destruction of swarm tightness) does not start before the relaxation time ${\tau}_{\mathrm{relax}}$.

## 4. Interactions between Collinear Colliding Swarms

- (i)
- ${\mathcal{S}}_{1}$-agents become leaders and thus are less influenced by (or possibly almost independent from) ${\mathcal{S}}_{2}$-agents. This implies that, with time, ${\mathcal{S}}_{1}$-agents will exhibit a net tendency to recover their nominal drifts (i.e., the drifts realized before the collision).
- (ii)
- ${\mathcal{S}}_{2}$-agents feel the presence of their leaders from ${\mathcal{S}}_{1}$ and, therefore, have a net tendency to increase their drifts.

- (a)
- Mutual capture of swarms. After collision, ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ aggregate into a global tight swarm ${\mathcal{S}}_{\mathrm{glob}}$. The average barycentric drift ${d}_{\mathrm{glob}}$ of ${\mathcal{S}}_{\mathrm{glob}}$ is larger than ${d}_{2}$. Thus, in this case, mutual interactions generate an increase in the average velocity of the global population.
- (b)
- Quasi-asymptotic freedom of swarms. After collision, and for asymptotic times, the swarms ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ evolve almost without interactions (quasi-free swarm evolutions). In the quasi-free regime, ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ recover their respective initial individual barycentric velocities ${d}_{1}$ and ${d}_{2}$.

#### 4.1. Colliding Swarms Driven by Hybrid Atlas Models

#### 4.2. Colliding Swarms Driven by Modified Hybrid Atlas (MHAM) Dynamics

**Figure 3.**Value of the critical threshold ${\rho}_{c}$, with regard to the swarm size N of the HAM, when ${N}_{1}={N}_{2}$. The asymptotic value $N\to \infty $ of ${\rho}_{c}=3$ is depicted in red.

**Figure 4.**Value of the critical threshold ${\rho}_{c}$, with regard to the proportion ${N}_{1}/N$ of fast agents in the HAM, when $N\to \infty $.

#### 4.3. Numerical Simulations

**Figure 5.**Final distributions ${P}_{1,2}(x,T)$ and average barycentric speeds ${v}_{1,2}\left(t\right)$ for one realization, in which both societies that start at ${x}_{0,\{1,2\}}=\mp 15$ achieve flocking. Here, $\rho =2.5<{\rho}_{c}$, $U=20>{U}_{c}=5.4$, $\sigma =1$ and ${N}_{1}={N}_{2}=250$ agents. (

**a**): distribution of the agents from ${\mathcal{S}}_{1}$ (blue) and ${\mathcal{S}}_{2}$ (red), at ending time $T=100$. (

**b**): average barycentric speed of each society, with respect to time.

**Figure 6.**Final distributions ${P}_{1,2}(x,T)$ and average barycentric speeds ${v}_{1,2}\left(t\right)$ for one realization, in which both societies that start at ${x}_{0,\{1,2\}}=\mp 15$ do not achieve flocking. Here, $\rho =6>{\rho}_{c}$, $U=20>{U}_{c}=2.26$, $\sigma =1$ and ${N}_{1}={N}_{2}=250$ agents. (

**a**): distribution of the agents from ${\mathcal{S}}_{1}$ (blue) and ${\mathcal{S}}_{2}$ (red), at ending time $T=100$. (

**b**): average barycentric speed of each society, with respect to time.

## 5. Conclusions and Perspectives

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Sartoretti, G.; Hongler, M.-O. Interacting Brownian Swarms: Some Analytical Results. *Entropy* **2016**, *18*, 27.
https://doi.org/10.3390/e18010027

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Sartoretti G, Hongler M-O. Interacting Brownian Swarms: Some Analytical Results. *Entropy*. 2016; 18(1):27.
https://doi.org/10.3390/e18010027

**Chicago/Turabian Style**

Sartoretti, Guillaume, and Max-Olivier Hongler. 2016. "Interacting Brownian Swarms: Some Analytical Results" *Entropy* 18, no. 1: 27.
https://doi.org/10.3390/e18010027